In an OFDM symbol the cyclic prefix is a repeat of the end of the symbol at the beginning. The purpose is to allow multipath to settle before the main data arrives at the receiver. The receiver is normally arranged to decode the signal after it has settled because this is when the frequencies become orthogonal to one another. The length of the cyclic prefix is often equal to the guard interval. After discovering the process for OFDM a cyclic prefix has been proposed for other modulations to improve the robustness to multipath. Mathematical Description

Principle

Cyclic prefix is often used in conjunction with modulation in order to retain sinusoids' properties in multipath channels. It is well known that sinusoidal signals are eigenfunctions of linear, and time-invariant systems. Therefore, if the channel is assumed to be linear and time-invariant, then a sinusoid of infinite duration would be an eigenfunction. However, in practice, this cannot be achieved, as real signals are always time-limited. So, to mimic the infinite behavior, prefixing the end of the symbol to the beginning makes the linear convolution of the channel appear as though it were circular convolution, and thus, preserve this property in the part of the symbol after the cyclic prefix.

Cyclic Prefix in Orthogonal Frequency Division Multiplexing

Cyclic Prefixes are used in OFDM in order to combat multipath by making channel estimation easy. As an example, consider an OFDM system which has N subcarriers. The message symbol can be written as:

\mathbf{d} = [d_0, d_1, \ldots d_{N - 1}]^T

The OFDM symbol is constructed by taking the Inverse Discrete Fourier Transform of the message symbol, followed by a cyclic prefixing. Let the symbol obtained by the Inverse Discrete Fourier Transform be denoted by

\mathbf{x_0} = [x[0], x[1], \ldots x[N - 1]]^T.

Prefixing it with a cyclic prefix of length L, the OFDM symbol obtained is:

\mathbf{x} = [x[N - L + 1], x[N - L + 2], \ldots x[N - 1], x[0], x[1], \ldots x[N - 1]]^T.

Assume that the channel is represented using

\mathbf{h} = [h_0, h_1, \ldots h_{L-1}]^T.

Then, after convolution with the channel, which happens as

y[m] = \sum_{l = 0}^{L - 1} h_l x[m - l] \quad 0 \le m \le N

which is circular convolution, as x[m - k] becomes x[(m - k)\mod N]. So, taking the Discrete Fourier Transform, we get

Y[k] = H[k]\cdot X[k].

where X[k] is the Discrete Fourier Transform of \mathbf{x}. Thus, the problem of channel estimation is simplified, as, once the values of \{H[k]\} are estimated, for the duration in which the channel does not vary significantly, merely multiplying the received demodulated symbols by the inverse of H[k] to obtain the actual symbols [d_0, d_1, \ldots d_{N - 1}]^T.

References

• Fundamentals of Wireless Communication, by David Tse and Pramod Viswanath, Cambridge University Press (2005).